Determine the minimum sample size required to be 95% confident that the sample mean waking time is within 10 minutes of the population mean waking time. Use the population standard deviation from Exercise 1.

Verified step by step guidance

1. Identify the formula for determining the minimum sample size for estimating a population mean: $n\; =\; (z\; *\; \sigma \; /\; E)^2$, where $n$ is the sample size, $z$ is the z-score corresponding to the confidence level, $\sigma$ is the population standard deviation, and $E$ is the margin of error.
2. Determine the z-score for a 95% confidence level. For a 95% confidence level, the z-score corresponds to the critical value where the cumulative probability is 0.975 (since 95% confidence leaves 2.5% in each tail).
3. Substitute the given margin of error, $E\; =\; 10$ minutes, into the formula. This represents the maximum allowable difference between the sample mean and the population mean.
4. Use the population standard deviation from Exercise 1, denoted as $\sigma$, and substitute it into the formula. Ensure the units are consistent (e.g., minutes).
5. Calculate the value of $n$ by squaring the result of $(z\; *\; \sigma \; /\; E)$. Round up to the nearest whole number, as sample size must be an integer.

Key Concepts

Sample Size Determination

Sample size determination is the process of calculating the number of observations or replicates needed in a statistical study to achieve a desired level of confidence and precision. In this context, it involves using the desired margin of error, confidence level, and population standard deviation to find the minimum sample size that ensures the sample mean is within a specified range of the population mean.

Confidence Level

The confidence level represents the probability that the confidence interval calculated from the sample data will contain the true population parameter. A 95% confidence level indicates that if the same sampling procedure were repeated multiple times, approximately 95% of the calculated intervals would capture the true population mean, providing a strong assurance of the reliability of the results.

Margin of Error

The margin of error is the range within which the true population parameter is expected to lie, given a certain level of confidence. In this scenario, a margin of error of 10 minutes means that the sample mean waking time should be within 10 minutes of the actual population mean. This concept is crucial for determining how precise the estimate needs to be and directly influences the required sample size.